Rigging stretches my model
Are you using an unapplied Skin modifier, right?

So the "real mesh", the one who receive the automatic weights from the armature it's actually made of a bunch of vertices, the ones that they are moved. The problem is all here: the vertices created by the skin modifier aren't affected by the armature because they don't really exist yet.

You can try to move the skin modifier to the bottom of the list, so it will occour after the armature deformation, but to get nice an smooth deformations, you'll have to provide more "real" vertices to the mesh. You can for example subdifide your skeleton or apply the skin modifier.